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在非线性误差增长理论框架下研究了混沌系统平均初始误差增长饱和特性 以及误差饱和值同系统可预报期限的关系.首先探索了Lorenz96系统中平均相对初始误差增长饱和规律, 发现平均相对初始误差增长饱和值同初始误差的自然对数存在简单的线性关系: 其二者自然对数之和为一常量,且该常量同初始误差无关.实验表明该结论对其他混沌系统也适用. 因此对给定混沌系统,在计算出和常数后可以外推得到任意固定初始误差的平均相对误差增长饱和值. 为进一步研究误差饱和值同可预报期限的关系,给出了平均绝对误差增长的定义. 理论分析表明混沌系统平均绝对误差增长也会达到饱和.其饱和值为常量, 与初始误差无关,混沌系统控制参数确定,饱和值就固定.依据上述研究, 最后给出一个定量计算可预报期限的模型Tp=1/∧ln(Es/δ0)+c, Es为绝对误差增长饱和值.实验研究表明对于复杂的高阶混沌系统,该预报期限模型都能较好地适用.The saturation property of mean growth of initial error and the relation between saturation value and predictability limit of chaos system are studied in a frame of the nonlinear error growth dynamics. Firstly, the saturation property of mean relative growth of initial error (RGIE) of Lorenz96 system is investigated. It is found that there exists a simple linear relationship between the logarithm of saturation value of mean RGIE and initial error. The sum of logarithms of the two is constant that is independent of the magnitude of the initial error. It is proven by experiment that this conclusion is suitable for other chaotic systems too. With this conclusion, once the constant sum has been determined, the saturation values of mean RGIE at any magnitude of initial error can be calculated easily. Furthermore, to make the study of the relation between error growth saturation and the predictability limits more convenient, just as the definition of the mean RGIE, a definition of the mean absolute growth of initial error (AGIE) is introduced and theoretical analysis reveals that the AGIE has a similar saturation property as RGIE. The saturation value of mean AGIE is constant, which means for a given chaos system, once the control parameters of the system has been determined, the saturation of AGIE is determined. Finally a model for calculating predictability limit quantitatively is given as follows: Tp=1/∧ln(Es/δ0)+c, where Es is the saturation value of mean AGIE. It is shown that this model can work with complicated and high dimension chaos system very well.
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Keywords:
- relative growth of initial error /
- nonlinear error growth dynamics /
- chaos /
- predictability limit
[1] Lorenz E N 1969 Tellus 21 289
[2] Eckmann J P, Ruelle D 1985 Rev. Mod. Phys. 57 617
[3] Ding R Q, Li J P 2007 Phys. Lett. A 364 396
[4] Li J P, Ding R Q, Chen B H 2006 Frontier and Prospect of Atmospheric Sciences at the Beginning of the 21th Century (Beijing: China Meteorology Press) p96 (in Chinese) [李建平, 丁瑞强, 陈宝花 2006 21世纪大气科学发展的回顾与展望 (北京:气象出版社) 第96页]
[5] Chen B H, Li J P, Ding R Q 2006 Sci. China D 49 1111
[6] Ding R Q, Li J P 2007 Chin. J. Atmos. Sci. 31 571 (in Chinese) [丁瑞强, 李建平 2007 大气科学 31 571]
[7] Ding R Q, Li J P 2008 Acta Phys. Sin. 57 7494 (in Chinese) [丁瑞强, 李建平 2008 57 7494]
[8] Wolf A, Swift J B, Swinney H L 1985 Physica D 16 285
[9] Sano M, Sawada Y 1985 Phys. Rev. Lett. 55 1082
[10] Li J P, Ding R Q 2011 Relationship between the Predictability Limit and Initial Error in Chaotic System Esteban Tlelo-Cuautle (Ed.) 39-50
[11] Lorenz E N 1995 Proceedings of a Seminar Held at ECMWF on Predictability (Reading: ECMWF) p1
[12] Diego Pazo, Ivan G Szendro 2008 Phys. Rev. E 78 16209
[13] Lorenz E N 1963 J. Atmos. Sci. 20 130
[14] Henon M 1976 Comm. Math. Phys. 50 69
[15] Rossler O E 1976 Phys. Lett. A 57 397
[16] Hu Y H 2009 Sci. Tech. Engng. 11 2856 (in Chinese) [胡杨慧 2009 科学技术与工程 11 2856]
[17] Orrell D, Smith L A 2001 Nonlin. Proc. Geo. 8 357
[18] Orrell D 2003 J. Atmos. Sci. 60 2219
[19] Lv J H, Lu J A, Chen S H 2005 (in Chinese) [吕金虎, 陆君安, 陈士华 2005 混沌时间序列分析及其应用(第二版) (武昌:武汉大学出版社) 第27页]
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[1] Lorenz E N 1969 Tellus 21 289
[2] Eckmann J P, Ruelle D 1985 Rev. Mod. Phys. 57 617
[3] Ding R Q, Li J P 2007 Phys. Lett. A 364 396
[4] Li J P, Ding R Q, Chen B H 2006 Frontier and Prospect of Atmospheric Sciences at the Beginning of the 21th Century (Beijing: China Meteorology Press) p96 (in Chinese) [李建平, 丁瑞强, 陈宝花 2006 21世纪大气科学发展的回顾与展望 (北京:气象出版社) 第96页]
[5] Chen B H, Li J P, Ding R Q 2006 Sci. China D 49 1111
[6] Ding R Q, Li J P 2007 Chin. J. Atmos. Sci. 31 571 (in Chinese) [丁瑞强, 李建平 2007 大气科学 31 571]
[7] Ding R Q, Li J P 2008 Acta Phys. Sin. 57 7494 (in Chinese) [丁瑞强, 李建平 2008 57 7494]
[8] Wolf A, Swift J B, Swinney H L 1985 Physica D 16 285
[9] Sano M, Sawada Y 1985 Phys. Rev. Lett. 55 1082
[10] Li J P, Ding R Q 2011 Relationship between the Predictability Limit and Initial Error in Chaotic System Esteban Tlelo-Cuautle (Ed.) 39-50
[11] Lorenz E N 1995 Proceedings of a Seminar Held at ECMWF on Predictability (Reading: ECMWF) p1
[12] Diego Pazo, Ivan G Szendro 2008 Phys. Rev. E 78 16209
[13] Lorenz E N 1963 J. Atmos. Sci. 20 130
[14] Henon M 1976 Comm. Math. Phys. 50 69
[15] Rossler O E 1976 Phys. Lett. A 57 397
[16] Hu Y H 2009 Sci. Tech. Engng. 11 2856 (in Chinese) [胡杨慧 2009 科学技术与工程 11 2856]
[17] Orrell D, Smith L A 2001 Nonlin. Proc. Geo. 8 357
[18] Orrell D 2003 J. Atmos. Sci. 60 2219
[19] Lv J H, Lu J A, Chen S H 2005 (in Chinese) [吕金虎, 陆君安, 陈士华 2005 混沌时间序列分析及其应用(第二版) (武昌:武汉大学出版社) 第27页]
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